4.3 Solving TrianglesSolve Each Triangle (all Sides And All Angles). Round Answers To The Nearest Tenth.1)$[ \begin Array}{c} \text{Given \ \angle A = 24.3^\circ, \ \angle B = 65.7^\circ, \ \angle C = 90^\circ, \ AB = 17, \ BC = 7, \ AC =

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Introduction

Solving triangles is a fundamental concept in geometry that involves finding the lengths of the sides and the measures of the angles of a triangle. In this article, we will focus on solving triangles with given information, including all sides and all angles. We will use the given information to find the missing sides and angles of the triangle, and we will round our answers to the nearest tenth.

Given Information

For the given triangle, we have the following information:

  • Angle A: 24.3°
  • Angle B: 65.7°
  • Angle C: 90° (right angle)
  • Side AB: 17 units
  • Side BC: 7 units
  • Side AC: unknown

Using the Law of Sines

The Law of Sines states that for any triangle with angles A, B, and C, and sides a, b, and c opposite to those angles, respectively, the following equation holds:

a / sin(A) = b / sin(B) = c / sin(C)

We can use this law to find the missing side AC.

Step 1: Find the measure of angle A

Since we know the measures of angles B and C, we can find the measure of angle A using the fact that the sum of the measures of the angles of a triangle is always 180°.

m∠A + m∠B + m∠C = 180° 24.3° + 65.7° + m∠A = 180° m∠A = 180° - 90° - 65.7° m∠A = 24.3°

Step 2: Find the length of side AC

Now that we know the measure of angle A, we can use the Law of Sines to find the length of side AC.

AC / sin(A) = AB / sin(B) AC / sin(24.3°) = 17 / sin(65.7°) AC = (17 / sin(65.7°)) * sin(24.3°) AC ≈ 10.5 units

Using the Law of Cosines

The Law of Cosines states that for any triangle with sides a, b, and c, and angle C opposite to side c, the following equation holds:

c² = a² + b² - 2ab * cos(C)

We can use this law to find the length of side AC.

Step 1: Find the length of side AC

Now that we know the length of side AB and the measure of angle C, we can use the Law of Cosines to find the length of side AC.

AC² = AB² + BC² - 2 * AB * BC * cos(C) AC² = 17² + 7² - 2 * 17 * 7 * cos(90°) AC² = 289 + 49 - 0 AC² = 338 AC ≈ √338 ≈ 18.4 units

Conclusion

In this article, we solved a triangle with given information, including all sides and all angles. We used the Law of Sines and the Law of Cosines to find the missing sides and angles of the triangle, and we rounded our answers to the nearest tenth. We found that the length of side AC is approximately 10.5 units using the Law of Sines and approximately 18.4 units using the Law of Cosines.

Discussion

The Law of Sines and the Law of Cosines are two fundamental concepts in geometry that are used to solve triangles. The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides and angles of a triangle. The Law of Cosines states that the square of the length of a side is equal to the sum of the squares of the other two sides minus twice the product of those two sides and the cosine of the angle between them.

Applications

Solving triangles has many practical applications in various fields, including:

  • Surveying: Solving triangles is used in surveying to determine the distances and angles between landmarks and reference points.
  • Navigation: Solving triangles is used in navigation to determine the position and course of a ship or aircraft.
  • Engineering: Solving triangles is used in engineering to determine the stresses and strains on structures and machines.
  • Computer Graphics: Solving triangles is used in computer graphics to create 3D models and animations.

References

  • Law of Sines: The Law of Sines states that for any triangle with angles A, B, and C, and sides a, b, and c opposite to those angles, respectively, the following equation holds: a / sin(A) = b / sin(B) = c / sin(C).
  • Law of Cosines: The Law of Cosines states that for any triangle with sides a, b, and c, and angle C opposite to side c, the following equation holds: c² = a² + b² - 2ab * cos(C).

Glossary

  • Angle: An angle is a measure of the amount of rotation between two lines or planes.
  • Side: A side is a line segment that connects two vertices of a polygon.
  • Triangle: A triangle is a polygon with three sides and three vertices.
  • Law of Sines: The Law of Sines is a mathematical formula that relates the lengths of the sides of a triangle to the sines of its angles.
  • Law of Cosines: The Law of Cosines is a mathematical formula that relates the lengths of the sides of a triangle to the cosine of one of its angles.
    Solving Triangles: A Comprehensive Guide =====================================================

Q&A: Solving Triangles

Q: What is the Law of Sines?

A: The Law of Sines is a mathematical formula that relates the lengths of the sides of a triangle to the sines of its angles. It states that for any triangle with angles A, B, and C, and sides a, b, and c opposite to those angles, respectively, the following equation holds: a / sin(A) = b / sin(B) = c / sin(C).

Q: What is the Law of Cosines?

A: The Law of Cosines is a mathematical formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. It states that for any triangle with sides a, b, and c, and angle C opposite to side c, the following equation holds: c² = a² + b² - 2ab * cos(C).

Q: How do I use the Law of Sines to solve a triangle?

A: To use the Law of Sines to solve a triangle, you need to know the measures of two angles and the length of one side. You can then use the formula a / sin(A) = b / sin(B) = c / sin(C) to find the lengths of the other sides.

Q: How do I use the Law of Cosines to solve a triangle?

A: To use the Law of Cosines to solve a triangle, you need to know the lengths of two sides and the measure of one angle. You can then use the formula c² = a² + b² - 2ab * cos(C) to find the length of the third side.

Q: What are some common mistakes to avoid when solving triangles?

A: Some common mistakes to avoid when solving triangles include:

  • Not checking for invalid triangles: Make sure that the triangle you are solving is valid, meaning that the sum of the measures of the angles is 180° and the lengths of the sides satisfy the triangle inequality.
  • Not using the correct formula: Make sure that you are using the correct formula for the Law of Sines or the Law of Cosines.
  • Not rounding correctly: Make sure that you are rounding your answers to the correct number of decimal places.

Q: What are some real-world applications of solving triangles?

A: Solving triangles has many real-world applications, including:

  • Surveying: Solving triangles is used in surveying to determine the distances and angles between landmarks and reference points.
  • Navigation: Solving triangles is used in navigation to determine the position and course of a ship or aircraft.
  • Engineering: Solving triangles is used in engineering to determine the stresses and strains on structures and machines.
  • Computer Graphics: Solving triangles is used in computer graphics to create 3D models and animations.

Q: How do I know which formula to use, the Law of Sines or the Law of Cosines?

A: To determine which formula to use, the Law of Sines or the Law of Cosines, you need to know the information that you have and the information that you are trying to find. If you know the measures of two angles and the length of one side, use the Law of Sines. If you know the lengths of two sides and the measure of one angle, use the Law of Cosines.

Q: Can I use the Law of Sines and the Law of Cosines together to solve a triangle?

A: Yes, you can use the Law of Sines and the Law of Cosines together to solve a triangle. For example, you can use the Law of Sines to find the length of one side and then use the Law of Cosines to find the length of another side.

Q: How do I check my work when solving triangles?

A: To check your work when solving triangles, you can use the following steps:

  • Check the triangle inequality: Make sure that the lengths of the sides satisfy the triangle inequality.
  • Check the sum of the angles: Make sure that the sum of the measures of the angles is 180°.
  • Check the formula: Make sure that you are using the correct formula for the Law of Sines or the Law of Cosines.
  • Check the rounding: Make sure that you are rounding your answers to the correct number of decimal places.